Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $k \neq 0$. $y = \dfrac{-10}{2(4k - 9)} \div \dfrac{k}{10(4k - 9)} $
Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{-10}{2(4k - 9)} \times \dfrac{10(4k - 9)}{k} $ When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ -10 \times 10(4k - 9) } { 2(4k - 9) \times k } $ $ y = \dfrac{-100(4k - 9)}{2k(4k - 9)} $ We can cancel the $4k - 9$ so long as $4k - 9 \neq 0$ Therefore $k \neq \dfrac{9}{4}$ $y = \dfrac{-100 \cancel{(4k - 9})}{2k \cancel{(4k - 9)}} = -\dfrac{100}{2k} = -\dfrac{50}{k} $